Theorem reseq1d 4941

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
reseq1d	|- ( ph -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq1 4936	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1364   |` cres 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-res 4671

This theorem is referenced by:  reseq12d  4943  fun2ssres  5297  funcnvres2  5329  funimaexg  5338  fresin  5432  offres  6187  tfrlemisucaccv  6378  tfrlemi1  6385  tfr1onlemsucaccv  6394  tfrcllemsucaccv  6407  freceq1  6445  freceq2  6446  fseq1p1m1  10160  setsresg  12656  setscom  12658  znle2  14140  dvcoapbr  14856  bj-charfundcALT  15301